Linear transporters are in particular also known as part of container treatment machines (EP 1 081 068 B1, EP 1 220 787 B1), with the containers being cycled, i.e. moved incrementally, along the transport path for their treatment at different treatment stations or treatment positions that succeed one another in the transport direction, while preferably held suspended by a container flange and in a multi-track container flow. Because of the cycled transport movement, the containers, which are held suspended, tend to swing or oscillate about axes oriented horizontal with and perpendicular to the transport direction. This impairs the reliability of the process. Among other things, there is also the danger that after the filling of the containers, filling material will spill out of the not-yet-sealed swinging containers, which then leads to a soiling and possibly also to an extremely undesirable contamination of the installation or device concerned with the spilt or splashed filling material, which then acts as a nutritive medium for bacteria.